MATLAB Examples

Arc-length control method (Crisfield, 1981 and Fafard & Massicotte, 1993)

Contents

Notation and references

The notation followed here and in the following MATLAB codes:

  • arc_length_Crisfield.m
  • arc_length_Crisfield_modified.m

conforms to that used by Fafard & Massicotte in the following reference:

Fafard, M. and Massicotte, B. (1993). ”Geometrical Interpretation of the Arc-Length Method.” Computers & Structures, 46(4), 603–615. This reference is denoted as [3] inside the text of the above codes.

Except for the above study, the following reference should be noted as well:

Crisfield, M. A. (1981). ”A Fast Incremental/Iterative Solution Procedure that Handles "Snap-Through".” Computers & Structures, 13(), 55–62. This reference is denoted as [4] inside the text of the above codes.

Algorithms implemented

  1. Arc length control method as described by Fafard & Massicotte (1993), after Crisfield (1981).
  2. Modified version of the above method which directs the search towards $$\mathrm{\lambda}=1$ , where $$\mathrm{\lambda}$ is the load factor.
help arc_length_Crisfield %1
help arc_length_Crisfield_modified %2
  Arc-length control method (Crisfield, 1981)
 
  Description
      The equation functn(#t#)=0 is solved for #t#, where
      #t#=[#u#;#lambda#], #u# is the unknown displacement vector and
      #lambda# is the unknown load factor. The method used is the
      arc-length method described by Crisfield (1981): "A Fast
      Incremental/Iterative Solution Procedure That Handles “Snap-Through”"
      with the following modifications:
      1.The capability to select between the cylindrical Crisfield method
      (original) or the spherical Crisfield method (described in [3] in the
      first paragraph after equation (35).
      2.The initial value of #lambda# is set equal to
      #lambda0#+#Deltalambdabar# instead of #Deltalambdabar# as is shown in
      Figure 8 in [3].
      The method is implemented according to the flow chart in Fig.8 and
      the procedure from equation (22) to equation (40) presented in [3].
 
  Required input arguments
      #functn# is the function handle defining the equation to be solved.
      The definition of #functn# must be of the type
      [#R#,#Q#,#K#]=functn(#t#) where #R# ([#dim# x 1]) is the out of
      balance force vector, #Q# ([#dim# x 1]) is the tangent load vector
      given by Q(a,lambda)=-d{R(a,lambda)}/d{lambda}, #K# ([#dim# x #dim#])
      is the tangent stiffness matrix given by
      K(a,lambda)=d{R(a,lambda)}/d{a} and #t# ([#dim#+1 x 1]) is the
      generalized unknown vector defined in the description section.
      #u0# ([#dim# x 1]) is the starting point of the solution.
 
  Optional input arguments
      #Crisver# (string) determines the version of the Crisfield method
      that will be used. It can take the values 'sph' (default) for the
      spherical Crisfield method or 'cyl' for the cylindrical Crisfield
      method (as published in [4]).
      #nmax# (scalar) is the maximum number of increments. Default value is
      30.
      #Deltalambdabar# (scalar) is the load increment at the first step.
      Default value is 1.
      #imax# (scalar) is the maximum number of iterations per increment.
      Default value is 12.
      #Id# (scalar) is the desired number of iterations per increment.
      Default value is 4.
      #tol# (scalar) is the tolerance for the convergence criterion.
      Default value is 5e-5.
      #KTup# (scalar) is the stiffness matrix updater (number of iterations
      after which the tangent stiffness matrix is updated). For #KTup# = 1
      the algorithm implemented is Full Arc-Length method. For #KTup# = Inf
      the algorithm implemented is Initial Stiffness Arc-Length method.
      Default value is 1.
      #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#).
      Default value is 1e-4.
 
  Output arguments
      #u# ([#dim# x #nmax#]) are the unknown displacements.
      #lambda# ([1 x #nmax#]) are the load factors (one per increment).
      #iter# ([1 x #nmax#]) is the number of iterations for each increment.
      #Aout# ([1 x #nmax#]) is the initial estimates of A (sign
      determinant) at each increment. The sign of A is positive along
      loading branches of the response curve (#lambda# increases) and is
      negative along unloading portions of the curve (#lambda# decreases).
      #DeltaSout# ([1 x #nmax#]) are the arc-length increments.
 
  Parents (calling functions)
      None.
 
  Children (called functions)
      None.
 
 __________________________________________________________________________
  Copyright (c) 09-Mar-2014
      George Papazafeiropoulos
      First Lieutenant, Infrastructure Engineer, Hellenic Air Force
      Civil Engineer, M.Sc., Ph.D. candidate, NTUA
      Email: gpapazafeiropoulos@yahoo.gr
      Website: http://users.ntua.gr/gpapazaf/
 
 

  Modified arc-length control method (Crisfield, 1981)
 
  Description
      The equation functn(#t#)=0 is solved for #t#, where
      #t#=[#u#;#lambda#], #u# is the unknown displacement vector and
      #lambda# is the unknown load factor. The method used is the
      arc-length method described by Crisfield (1981): "A Fast
      Incremental/Iterative Solution Procedure That Handles “Snap-Through”"
      with the following modifications:
      1.The capability to select between the cylindrical Crisfield method
      (original) or the spherical Crisfield method (described in [3] in the
      first paragraph after equation (35).
      2.The initial value of #lambda# is set equal to
      #lambda0#+#Deltalambdabar# instead of #Deltalambdabar# as is shown in
      Figure (8) in [3].
      3.The solution procedure is directed towards #lambda#=1, where
      #lambda# is the load factor.
      The method is implemented according to the flow chart in Fig.8 and
      the procedure from equation (22) to equation (40) presented in [3].
 
  Required input arguments
      #functn# is the function handle defining the equation to be solved.
      The definition of #functn# must be of the type
      [#R#,#Q#,#K#]=functn(#t#) where #R# ([#dim# x 1]) is the out of
      balance force vector, #Q# ([#dim# x 1]) is the tangent load vector
      given by Q(a,lambda)=-d{R(a,lambda)}/d{lambda}, #K# ([#dim# x #dim#])
      is the tangent stiffness matrix given by
      K(a,lambda)=d{R(a,lambda)}/d{a} and #t# ([#dim#+1 x 1]) is the
      generalized unknown vector defined in the description section.
      #u0# ([#dim# x 1]) is the starting point of the solution.
 
  Optional input arguments
      #Crisver# (string) determines the version of the Crisfield method
      that will be used. It can take the values 'sph' (default) for the
      spherical Crisfield method or 'cyl' for the cylindrical Crisfield
      method (as published in [4]).
      #nmax# (scalar) is the maximum number of increments. Default value is
      30.
      #Deltalambdabar# (scalar) is the load increment at the first step.
      Default value is 1.
      #imax# (scalar) is the maximum number of iterations per increment.
      Default value is 12.
      #Id# (scalar) is the desired number of iterations per increment.
      Default value is 4.
      #tol# (scalar) is the tolerance for the convergence criterion.
      Default value is 5e-5.
      #KTup# (scalar) is the stiffness matrix updater (number of iterations
      after which the tangent stiffness matrix is updated). For #KTup# = 1
      the algorithm implemented is Full Arc-Length method. For #KTup# = Inf
      the algorithm implemented is Initial Stiffness Arc-Length method.
      Default value is 1.
      #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#).
      Default value is 1e-4.
 
  Output arguments
      #u# ([#dim# x #nmax#]) are the unknown displacements.
      #lambda# ([1 x #nmax#]) are the load factors (one per increment).
      #iter# ([1 x #nmax#]) is the number of iterations for each increment.
      #Aout# ([1 x #nmax#]) is the initial estimates of A (sign
      determinant) at each increment. The sign of A is positive along
      loading branches of the response curve (#lambda# increases) and is
      negative along unloading portions of the curve (#lambda# decreases).
      #DeltaSout# ([1 x #nmax#]) are the arc-length increments.
 
  Parents (calling functions)
      None.
 
  Children (called functions)
      None.
 
 __________________________________________________________________________
  Copyright (c) 09-Mar-2014
      George Papazafeiropoulos
      First Lieutenant, Infrastructure Engineer, Hellenic Air Force
      Civil Engineer, M.Sc., Ph.D. candidate, NTUA
      Email: gpapazafeiropoulos@yahoo.gr
      Website: http://users.ntua.gr/gpapazaf/
 
 

Crisfield's arc-length method is described also in the material distributed to students of the Analysis and Design of Earthquake Resistant Structures postgraduate course of the School of Civil Engineering, NTUA, at the subject "Nonlinear Finite Elements" with Prof. M. Papadrakakis as course instructor. This version includes some improvements compared to the actual course material.

Equations solved

The following equations are solved for $$\mathrm{x_i}$ and $$\mathrm{\lambda}$

$$x^3 - \frac{57\, x^2}{8} + \frac{51\, x}{4} = 5\, \mathrm{\lambda} \ \
\ \ \ \ (1)$$

$$\left[\begin{array}{c} {\mathrm{x_1}}^2 + {\mathrm{x_2}}^2 - 49\\
\mathrm{x_1}\, \mathrm{x_2} - 24 \end{array}\right] =
\left[\begin{array}{c} 1\\ 1 \end{array}\right] \, \mathrm{\lambda} \ \ \
\ \ \ (2)$$

Function definitions

Two functions are utilized for the arc-length procedure:

The first function ($f_1$, defined in the file function2.m ), needed to solve equation (1) is a cubic polynomial with the following properties:

  • Function value:

$$f_1\left( x \right) = x^3 - \frac{57\, x^2}{8} + \frac{51\, x}{4}$$

  • Function jacobian (derivative):

$$J_1\left( x \right) = 3\, x^2 - \frac{57\, x}{4} + \frac{51}{4} $$

  • Passes through the origin:

$$f_1\left( 0 \right) = 0 $$

The second function ($f_2$, defined in the file function1.m ), needed to solve equation (2) is a nonlinear smooth function with the following properties:

  • Function value:

$$f_2\left(\left[\begin{array}{c} \mathrm{x_1}\\ \mathrm{x_2}
\end{array}\right]\right) = \left[\begin{array}{c} {\mathrm{x_1}}^2 +
{\mathrm{x_2}}^2 - 49\\ \mathrm{x_1}\, \mathrm{x_2} - 24
\end{array}\right]$$

  • Function jacobian:

$$J_2\left(\left[\begin{array}{c} \mathrm{x_1}\\ \mathrm{x_2}
\end{array}\right]\right) = \left[\begin{array}{cc} 2\, \mathrm{x_1} &
2\, \mathrm{x_2}\\ \mathrm{x_2} & \mathrm{x_1} \end{array}\right] $$

Function coding

  • For function $f_1$:
function [R,Q,K]=function2(t)
a=t(1:end-1);
lambda=t(end);
f1=a^3-57/8*a^2+51/4*a;
Rint=f1;
Rext=lambda*5;
% Out of balance force column vector (1-by-1)
R=Rint-Rext;
% Tangent force column vector (1-by-1)
Q=5;
% Jacobian matrix (1-by-1)
K=3*a^2-57/4*a+51/4;
end
  • For function $f_2$:
function [R,Q,K]=function1(t)
a=t(1:end-1);
lambda=t(end);
f1=a(1)^2+a(2)^2-49;
f2=a(1)*a(2)-24;
Rint=[f1;f2];
Rext=lambda*[1;1];
% Out of balance force column vector (2-by-1)
R=Rint-Rext;
% Tangent force column vector (2-by-1)
Q=[1;1];
% Jacobian matrix (2-by-2)
K=[2*a(1), 2*a(2);
    a(2), a(1)];
end

Initial definitions

In the subsequent code the following initial definitions are made (in the order presented below):

  1. Define function $f_1$
  2. Define function $f_2$
  3. Set starting point ($u_0$) for solution of equation (1)
  4. Set starting point ($u_0$) for solution of equation (2)
  5. Define the version of the Crisfield method to be used (cylindrical is selected)
  6. Set number of increments desired
  7. Set initial value of $$\mathrm{\lambda_0}$
  8. Set initial value of $\overline{\mathrm{\Delta\lambda}}$
  9. Set maximum number of iterations permitted per increment
  10. Set number of iterations desired to each converged point ($$I^{d}$)
  11. Set tolerance for convergence.
  12. Set the number of iterations every which the stiffness matrix of the problem is updated. KTup=1 corresponds to the full arc length method
  13. Set the tolerance for determining if the stiffness matrix is singular (this is true if its determinant is below dettol)
functn1=@function2; %1
functn2=@function1; %2
u01=0.1; %3
u02=[4;6]; %4
Crisver='cyl'; %5
nmax=30; %6
lambda0=0; %7
Deltalambdabar=0.4; %8
imax=20; %9
Id=1; %10
tol=5e-5; %11
KTup=1; %12
dettol=1e-4; %13

Applications

  1. Default application of the arc length control method as described by Crisfield (1981) to solve equation (1)
  2. Default application of the modified version of the Crisfield (1981) arc length control method to solve equation (1)
  3. Non-default application of the arc length control method as described by Crisfield (1981) to solve equation (1)
  4. Non-default application of the modified version of the Crisfield (1981) arc length control method to solve equation (1)
  5. Default application of the arc length control method as described by Crisfield (1981) to solve equation (2)
  6. Default application of the modified version of the Crisfield (1981) arc length control method to solve equation (2)
  7. Non-default application of the arc length control method as described by Crisfield (1981) to solve equation (2)
  8. Non-default application of the modified version of the Crisfield (1981) arc length control method to solve equation (2)
[u1,lambda1,iter1,Aout1,DeltaSout1] = arc_length_Crisfield(functn1,u01); %1
Result1=[u1',lambda1',iter1',Aout1',DeltaSout1'] %1
[u2,lambda2,iter2,Aout2,DeltaSout2] = arc_length_Crisfield_modified(functn1,u01); %2
Result2=[u2',lambda2',iter2',Aout2',DeltaSout2'] %2
[u3,lambda3,iter3,Aout3,DeltaSout3] = arc_length_Crisfield(functn1,u01,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %3
Result3=[u3',lambda3',iter3',Aout3',DeltaSout3'] %3
[u4,lambda4,iter4,Aout4,DeltaSout4] = arc_length_Crisfield_modified(functn1,u01,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %4
Result4=[u4',lambda4',iter4',Aout4',DeltaSout4'] %4
[u5,lambda5,iter5,Aout5,DeltaSout5] = arc_length_Crisfield(functn2,u02); %5
Result5=[u5',lambda5',iter5',Aout5',DeltaSout5'] %5
[u6,lambda6,iter6,Aout6,DeltaSout6] = arc_length_Crisfield_modified(functn2,u02); %6
Result6=[u6',lambda6',iter6',Aout6',DeltaSout6'] %6
[u7,lambda7,iter7,Aout7,DeltaSout7] = arc_length_Crisfield(functn2,u02,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %7
Result7=[u7',lambda7',iter7',Aout7',DeltaSout7'] %7
[u8,lambda8,iter8,Aout8,DeltaSout8] = arc_length_Crisfield_modified(functn2,u02,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %8
Result8=[u8',lambda8',iter8',Aout8',DeltaSout8'] %8
Result1 =

   1.0e+12 *

    0.0000    0.0000    0.0000         0    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000   -0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0000    0.0000    0.0000    0.0000
    0.0000    0.0001    0.0000    0.0000    0.0000
    0.0000    0.0001    0.0000    0.0000    0.0000
    0.0000    0.0003    0.0000    0.0000    0.0000
    0.0000    0.0007    0.0000    0.0000    0.0000
    0.0000    0.0017    0.0000    0.0000    0.0000
    0.0000    0.0042    0.0000    0.0000    0.0000
    0.0000    0.0099    0.0000    0.0000    0.0000
    0.0000    0.0234    0.0000    0.0000    0.0000
    0.0000    0.0556    0.0000    0.0000    0.0000
    0.0000    0.1318    0.0000    0.0000    0.0000
    0.0000    0.3128    0.0000    0.0000    0.0000
    0.0000    0.7418    0.0000    0.0000    0.0000
    0.0000    1.7589    0.0000    0.0000    0.0000
    0.0000    4.1706    0.0000    0.0000    0.0000


Result2 =

   -0.9927    1.3440   12.0000         0    1.0927
   -0.6284   -2.2149    3.0000   -0.1830    0.3642
   -0.1428   -0.3938    3.0000    0.0796    0.4856
    0.5047    3.7003   12.0000    0.1636    0.6475
    0.7205    1.3895   12.0000    0.5121    0.2158
    0.6486    1.1090    3.0000    0.2671    0.0719
    0.5526    1.0078    3.0000   -0.0754    0.0959
    0.6806    1.2117   12.0000   -0.0828    0.1279
    0.6379    1.0987    3.0000    0.1440    0.0426
    0.5811    1.0398    3.0000   -0.0437    0.0568
    0.6569    1.1419   12.0000   -0.0518    0.0758
    0.6316    1.0925    3.0000    0.0809    0.0253
    0.5979    1.0580    3.0000   -0.0255    0.0337
    0.5530    1.0082    3.0000   -0.0318    0.0449
    0.6129    1.0895   12.0000   -0.0388    0.0599
    0.5929    1.0527    3.0000    0.0582    0.0200
    0.5663    1.0234    3.0000   -0.0186    0.0266
    0.6018    1.0676   12.0000   -0.0236    0.0355
    0.5900    1.0495    3.0000    0.0337    0.0118
    0.5742    1.0322    2.0000   -0.0110    0.0158
    0.6057    1.0706   12.0000   -0.0142    0.0315
    0.5952    1.0551    3.0000    0.0302    0.0105
    0.5812    1.0400    2.0000   -0.0099    0.0140
    0.5532    1.0084    3.0000   -0.0128    0.0280
    0.5906    1.0563   12.0000   -0.0242    0.0374
    0.5781    1.0365    3.0000    0.0347    0.0125
    0.5615    1.0179    2.0000   -0.0113    0.0166
    0.5947    1.0594   12.0000   -0.0146    0.0332
    0.5836    1.0426    3.0000    0.0311    0.0111
    0.5689    1.0263    2.0000   -0.0102    0.0148


Result3 =

    0.2761    0.5997    2.0000         0    0.1761
    0.3642    0.7494    2.0000    0.0974    0.0881
    0.4082    0.8171    2.0000    0.0553    0.0440
    0.4303    0.8493    2.0000    0.0296    0.0220
    0.4413    0.8649    2.0000    0.0153    0.0110
    0.4468    0.8727    1.0000    0.0078    0.0055
    0.4523    0.8803    1.0000    0.0039    0.0055
    0.4578    0.8879    1.0000    0.0040    0.0055
    0.4633    0.8954    1.0000    0.0040    0.0055
    0.4688    0.9028    1.0000    0.0041    0.0055
    0.4743    0.9102    1.0000    0.0041    0.0055
    0.4798    0.9175    1.0000    0.0041    0.0055
    0.4853    0.9248    1.0000    0.0042    0.0055
    0.4908    0.9319    1.0000    0.0042    0.0055
    0.4963    0.9390    1.0000    0.0042    0.0055
    0.5018    0.9460    1.0000    0.0043    0.0055
    0.5073    0.9530    1.0000    0.0043    0.0055
    0.5128    0.9599    1.0000    0.0044    0.0055
    0.5183    0.9667    1.0000    0.0044    0.0055
    0.5238    0.9735    1.0000    0.0045    0.0055
    0.5293    0.9802    1.0000    0.0045    0.0055
    0.5348    0.9868    1.0000    0.0046    0.0055
    0.5403    0.9934    1.0000    0.0046    0.0055
    0.5458    0.9998    1.0000    0.0046    0.0055
    0.5513    1.0063    1.0000    0.0047    0.0055
    0.5568    1.0126    1.0000    0.0047    0.0055
    0.5624    1.0189    1.0000    0.0048    0.0055
    0.5679    1.0251    1.0000    0.0048    0.0055
    0.5734    1.0313    1.0000    0.0049    0.0055
    0.5789    1.0374    1.0000    0.0049    0.0055


Result4 =

    0.2761    0.5997    2.0000         0    0.1761
    0.3642    0.7494    2.0000    0.0974    0.0881
    0.4082    0.8171    2.0000    0.0553    0.0440
    0.4303    0.8493    2.0000    0.0296    0.0220
    0.4413    0.8649    2.0000    0.0153    0.0110
    0.4468    0.8727    1.0000    0.0078    0.0055
    0.4523    0.8803    1.0000    0.0039    0.0055
    0.4578    0.8879    1.0000    0.0040    0.0055
    0.4633    0.8954    1.0000    0.0040    0.0055
    0.4688    0.9028    1.0000    0.0041    0.0055
    0.4743    0.9102    1.0000    0.0041    0.0055
    0.4798    0.9175    1.0000    0.0041    0.0055
    0.4853    0.9248    1.0000    0.0042    0.0055
    0.4908    0.9319    1.0000    0.0042    0.0055
    0.4963    0.9390    1.0000    0.0042    0.0055
    0.5018    0.9460    1.0000    0.0043    0.0055
    0.5073    0.9530    1.0000    0.0043    0.0055
    0.5128    0.9599    1.0000    0.0044    0.0055
    0.5183    0.9667    1.0000    0.0044    0.0055
    0.5238    0.9735    1.0000    0.0045    0.0055
    0.5293    0.9802    1.0000    0.0045    0.0055
    0.5348    0.9868    1.0000    0.0046    0.0055
    0.5403    0.9934    1.0000    0.0046    0.0055
    0.5458    0.9998    1.0000    0.0046    0.0055
    0.5403    0.9935   20.0000    0.0047    0.0055
    0.5406    0.9937    2.0000   -0.0046    0.0003
    0.5407    0.9938    1.0000    0.0002    0.0001
    0.5409    0.9940    1.0000    0.0001    0.0001
    0.5410    0.9942    1.0000    0.0001    0.0001
    0.5412    0.9943    1.0000    0.0001    0.0001


Result5 =

    4.6983    5.2551    0.6900    4.0000         0    1.0210
    5.3835    4.4982    0.2161    4.0000    0.6445    1.0210
    5.7208    3.5345   -3.7801    4.0000   -0.4127    1.0210
    5.7581    2.5141   -9.5235    4.0000   -0.1995    1.0210
    5.5806    1.5086  -15.5810    4.0000   -0.1706    1.0210
    5.2493    0.5428  -21.1505    4.0000   -0.1760    1.0210
    4.8020   -0.3750  -25.8006    4.0000   -0.2017    1.0210
    4.2622   -1.2417  -29.2922    3.0000   -0.2540    1.0210
    3.4237   -2.3142  -31.9230    3.0000   -0.3641    1.3614
    2.1272   -3.5846  -31.6253    3.0000   -1.3020    1.8152
    0.1211   -4.9384  -24.5978    4.0000    1.3221    2.4202
   -2.1700   -5.7184  -11.5910    4.0000    0.5553    2.4202
   -4.5614   -5.3459    0.3846    5.0000    0.3984    2.4202
   -5.6740   -3.7613   -2.6585    4.0000    1.0337    1.9362
   -5.6562   -1.8252  -13.6764    4.0000   -0.3901    1.9362
   -5.0015   -0.0031  -23.9847    5.0000   -0.3211    1.9362
   -4.2041    1.3249  -29.5699    4.0000   -0.4296    1.5489
   -3.2301    2.5292  -32.1696    4.0000   -0.5776    1.5489
   -2.1100    3.5991  -31.5942    3.0000   -2.3411    1.5489
   -0.4153    4.7794  -25.9848    4.0000    1.1052    2.0653
    1.4903    5.5757  -15.6906    4.0000    0.5189    2.0653
    3.5506    5.7180   -3.6975    4.0000    0.3547    2.0653
    5.3041    4.6269    0.5413    5.0000    0.3995    2.0653
    5.7707    3.0419   -6.4462    4.0000   -1.0001    1.6522
    5.5520    1.4042  -16.2037    4.0000   -0.2831    1.6522
    4.9346   -0.1283  -24.6330    4.0000   -0.2857    1.6522
    4.0574   -1.5284  -30.2013    4.0000   -0.3807    1.6522
    2.9861   -2.7862  -32.3200    4.0000   -0.6988    1.6522
    1.7539   -3.8868  -30.8169    3.0000   -8.7454    1.6522
   -0.1134   -5.0557  -23.4269    4.0000    0.8261    2.2029


Result6 =

    5.0110    5.8573    6.2854   12.0000         0    1.0210
    4.6371    5.2973    0.5642   10.0000    0.4007    0.3403
    4.7482    5.2186    0.7788    3.0000   -0.0000    0.1361
    4.8890    5.1040    0.9538    3.0000    0.1033    0.1815
    4.6995    5.2545    0.6920   12.0000    0.2989    0.2420
    4.7642    5.2064    0.8045    3.0000   -0.1561    0.0807
    4.8482    5.1391    0.9153    3.0000    0.0651    0.1076
    4.7470    5.2408    0.9182   12.0000    0.1312    0.1434
    4.7723    5.2002    0.8169    3.0000   -0.1027    0.0478
    4.8222    5.1606    0.8855    3.0000    0.0373    0.0637
    4.8871    5.1057    0.9522    3.0000    0.0669    0.0850
    4.8075    5.1863    0.9582   12.0000    0.1377    0.1133
    4.8288    5.1552    0.8935    3.0000   -0.1057    0.0378
    4.8674    5.1228    0.9348    3.0000    0.0395    0.0504
    4.9178    5.0784    0.9742    3.0000    0.0699    0.0672
    4.8550    5.1422    0.9813   12.0000    0.1480    0.0895
    4.8727    5.1182    0.9397    3.0000   -0.1101    0.0298
    4.9027    5.0920    0.9642    3.0000    0.0420    0.0398
    4.9418    5.0562    0.9869    3.0000    0.0744    0.0531
    4.8924    5.1067    0.9939   12.0000    0.1641    0.0707
    4.9069    5.0882    0.9671    3.0000   -0.1166    0.0236
    4.9302    5.0670    0.9813    2.0000    0.0454    0.0314
    4.8856    5.1114    0.9805   12.0000    0.0813    0.0629
    4.8990    5.0953    0.9615    3.0000   -0.0985    0.0210
    4.9197    5.0766    0.9754    2.0000    0.0374    0.0279
    4.9606    5.0385    0.9939    2.0000    0.0631    0.0559
    4.8780    5.1137    0.9444   12.0000    0.2540    0.1118
    4.9059    5.0891    0.9665    3.0000   -0.1680    0.0373
    4.9426    5.0555    0.9872    3.0000    0.0720    0.0497
    4.8961    5.1027    0.9923   12.0000    0.1556    0.0662


Result7 =

    3.8946    5.6384   -2.0406    5.0000         0    0.0825
    3.9105    5.6337   -1.9696    2.0000    0.0000    0.0165
    3.9184    5.6314   -1.9344    1.0000    0.0038    0.0082
    3.9263    5.6290   -1.8993    1.0000    0.0019    0.0082
    3.9341    5.6266   -1.8643    1.0000    0.0019    0.0082
    3.9420    5.6241   -1.8295    1.0000    0.0020    0.0082
    3.9499    5.6217   -1.7949    1.0000    0.0020    0.0082
    3.9578    5.6192   -1.7605    1.0000    0.0020    0.0082
    3.9656    5.6167   -1.7262    1.0000    0.0020    0.0082
    3.9735    5.6142   -1.6920    1.0000    0.0020    0.0082
    3.9813    5.6117   -1.6581    2.0000    0.0020    0.0082
    3.9852    5.6104   -1.6412    1.0000    0.0020    0.0041
    3.9892    5.6091   -1.6243    1.0000    0.0010    0.0041
    3.9931    5.6078   -1.6074    1.0000    0.0010    0.0041
    3.9970    5.6065   -1.5907    1.0000    0.0010    0.0041
    4.0009    5.6052   -1.5739    1.0000    0.0010    0.0041
    4.0048    5.6039   -1.5572    1.0000    0.0010    0.0041
    4.0087    5.6026   -1.5405    1.0000    0.0010    0.0041
    4.0126    5.6013   -1.5239    1.0000    0.0010    0.0041
    4.0165    5.6000   -1.5073    1.0000    0.0010    0.0041
    4.0204    5.5987   -1.4908    1.0000    0.0010    0.0041
    4.0244    5.5973   -1.4743    1.0000    0.0010    0.0041
    4.0283    5.5960   -1.4579    1.0000    0.0010    0.0041
    4.0322    5.5947   -1.4415    1.0000    0.0010    0.0041
    4.0360    5.5933   -1.4251    1.0000    0.0010    0.0041
    4.0399    5.5920   -1.4088    1.0000    0.0010    0.0041
    4.0438    5.5906   -1.3925    1.0000    0.0010    0.0041
    4.0477    5.5892   -1.3763    1.0000    0.0010    0.0041
    4.0516    5.5879   -1.3601    1.0000    0.0010    0.0041
    4.0555    5.5865   -1.3439    1.0000    0.0011    0.0041


Result8 =

    3.8946    5.6384   -2.0406    5.0000         0    0.0825
    3.9105    5.6337   -1.9696    2.0000    0.0000    0.0165
    3.9184    5.6314   -1.9344    1.0000    0.0038    0.0082
    3.9263    5.6290   -1.8993    1.0000    0.0019    0.0082
    3.9341    5.6266   -1.8643    1.0000    0.0019    0.0082
    3.9420    5.6241   -1.8295    1.0000    0.0020    0.0082
    3.9499    5.6217   -1.7949    1.0000    0.0020    0.0082
    3.9578    5.6192   -1.7605    1.0000    0.0020    0.0082
    3.9656    5.6167   -1.7262    1.0000    0.0020    0.0082
    3.9735    5.6142   -1.6920    1.0000    0.0020    0.0082
    3.9813    5.6117   -1.6581    2.0000    0.0020    0.0082
    3.9852    5.6104   -1.6412    1.0000    0.0020    0.0041
    3.9892    5.6091   -1.6243    1.0000    0.0010    0.0041
    3.9931    5.6078   -1.6074    1.0000    0.0010    0.0041
    3.9970    5.6065   -1.5907    1.0000    0.0010    0.0041
    4.0009    5.6052   -1.5739    1.0000    0.0010    0.0041
    4.0048    5.6039   -1.5572    1.0000    0.0010    0.0041
    4.0087    5.6026   -1.5405    1.0000    0.0010    0.0041
    4.0126    5.6013   -1.5239    1.0000    0.0010    0.0041
    4.0165    5.6000   -1.5073    1.0000    0.0010    0.0041
    4.0204    5.5987   -1.4908    1.0000    0.0010    0.0041
    4.0244    5.5973   -1.4743    1.0000    0.0010    0.0041
    4.0283    5.5960   -1.4579    1.0000    0.0010    0.0041
    4.0322    5.5947   -1.4415    1.0000    0.0010    0.0041
    4.0360    5.5933   -1.4251    1.0000    0.0010    0.0041
    4.0399    5.5920   -1.4088    1.0000    0.0010    0.0041
    4.0438    5.5906   -1.3925    1.0000    0.0010    0.0041
    4.0477    5.5892   -1.3763    1.0000    0.0010    0.0041
    4.0516    5.5879   -1.3601    1.0000    0.0010    0.0041
    4.0555    5.5865   -1.3439    1.0000    0.0011    0.0041

Copyright

Copyright (c) 09-Mar-2014 by George Papazafeiropoulos